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A goal-oriented adaptive procedure for the quasi-continuum method with cluster approximation

Arash Memarnahavandi (Institutionen för tillämpad mekanik, Material- och beräkningsmekanik) ; Fredrik Larsson (Institutionen för tillämpad mekanik, Material- och beräkningsmekanik) ; Kenneth Runesson (Institutionen för tillämpad mekanik, Material- och beräkningsmekanik)
Computational Mechanics (0178-7675). Vol. 55 (2015), 4, p. 617-642.
[Artikel, refereegranskad vetenskaplig]

We present a strategy for adaptive error control for the quasi-continuum (QC) method applied to molecular statics problems. The QC-method is introduced in two steps: Firstly, introducing QC-interpolation while accounting for the exact summation of all the bond-energies, we compute goal-oriented error estimators in a straight-forward fashion based on the pertinent adjoint (dual) problem. Secondly, for large QC-elements the bond energy and its derivatives are typically computed using an appropriate discrete quadrature using cluster approximations, which introduces a model error. The combined error is estimated approximately based on the same dual problem in conjunction with a hierarchical strategy for approximating the residual. As a model problem, we carry out atomistic-to-continuum homogenization of a graphene monolayer, where the Carbon-Carbon energy bonds are modeled via the Tersoff-Brenner potential, which involves next-nearest neighbor couplings. In particular, we are interested in computing the representative response for an imperfect lattice. Within the goal-oriented framework it becomes natural to choose the macro-scale (continuum) stress as the "quantity of interest". Two different formulations are adopted: The Basic formulation and the Global formulation. The presented numerical investigation shows the accuracy and robustness of the proposed error estimator and the pertinent adaptive algorithm.

Nyckelord: Goal-oriented adaptivity, Quasi-continuum, Cluster approximation, Graphene, NONLINEAR MECHANICS, GRAIN-BOUNDARIES, LAYER GRAPHENE, ERROR CONTROL, ATOMIC-SCALE, STRENGTH, NANOINDENTATION, DEFORMATION, FORMULATION, SHEETS, Mathematics, Interdisciplinary Applications, Mechanics

Denna post skapades 2015-05-26. Senast ändrad 2015-09-25.
CPL Pubid: 217562


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